We note first that division by zero is not allowed and that for this reason \(6x+7\) may not be equal to zero and that therefore \(x=-{{7}\over{6}}\) is not a solution.

We now distinguish two cases, namely \(6x+7>0\) and \(6x+7<0\).
In both cases we multiply the inequality on both sides by \(6x+7\) because we then get a linear inequality, for which we know there is a solution method.

Suppose \(6x+7>0\), i.e. \(x> -{{7}\over{6}}\). Then we get \(9<-4(6x+7)\).
When we move everything with \(x\) to the left and all constant terms to the right, we get \(24x<-37\).
Then, dvision by the coefficient of \(x\)gives \(x < -{{37}\over{24}}\).
So we have the following system of inequalities: \(x> -{{7}\over{6}}\,\wedge\; x < -{{37}\over{24}}\)
and this simplifies to \(\text{an empty solution set}\).

Suppose \(6x+7<0\), i.e. \(x< -{{7}\over{6}}\). Then we get \(9>-4(6x+7)\).
When we move everything with \(x\) to the left and all constant terms to the right, we get \(24x>-37\).
Then, division by the coefficient of \(x\) gives \(x > -{{37}\over{24}}\).
So we have the following system of inequalities: \(x< -{{7}\over{6}}\,\wedge\; x > -{{37}\over{24}}\)
and this simplifies to \(-{{37}\over{24}}\lt x\lt -{{7}\over{6}}\).

The solution of the original inequality is \(-{{37}\over{24}}\lt x\lt -{{7}\over{6}}\).